Open Access

Existence of Solutions for Hyperbolic System of Second Order Outside a Domain

Journal of Inequalities and Applications20092009:489061

DOI: 10.1155/2009/489061

Received: 27 June 2008

Accepted: 29 April 2009

Published: 5 May 2009


We study the mixed initial-boundary value problem for hyperbolic system of second order outside a closed domain. The existence of solutions to this problem is proved and the estimate for the regularity of solutions is given. The application of the existence theorem to elastrodynamics is discussed.

1. Introduction

This paper is concerned with the exterior problem for hyperbolic system of second order. Let be a closed domain with smooth boundary in and let the origin belong to . Consider the following exterior problem for the hyperbolic system of second order:

where and . We assume that satisfies

for all symmetric matrixes , where , .

Let . The system (1.1) can be written as an evolution system in the form

Ikawa considered in [1] the mixed problem of a hyperbolic equation of second-order. The existence theorem is known for the obstacle free problem in [2]. Dafermos and Hrusa proved in [3] the local existence of the Dirichlet problem for the hyperbolic system inside a domain by energy method.

In this paper, we deal with the exterior problem for the second order hyperbolic system. In Section 2, we show the existence of the exterior problem for the problem (1.1) by the semigroup theory. In Section 3, we prove the regularity for the solutions of the exterior problem (1.1) and give the estimate for the regularity of solutions. In Section 4, we discuss the application of the existence theorem to elastrodynamics.

2. Existence of the Exterior Problem for Hyperbolic System of Second Order

Note that with the inner product

By (1.2) and Korn inequality (cf. [4, 5]), we have

Lemma 2.1.

For some , we have
Then is a Hilbert space with the inner product defined as above. We define the operator (without loss of generality, we still write this operator as ) in by

where It is obvious that is a densely defined operator.

Lemma 2.2.

There exists a constant such that for any ,



Let .

Corollary 2.3.

For all real such that , the estimate

holds for any .


By (2.4),

The estimate of the resolvent operator is the following.

Lemma 2.4.

There exists a constant such that for all real and ,
is a bijective mapping. Moreover, we have


Consider the system

where .

The substitution of the first relation
in the second of (2.11) gives
By the well-known variation method, there exists a solution of the elliptic system (2.13) for any . Defining by (2.12), we have a solution

of (2.10). Therefore, is a surjection.

From (2.6), it follows that the existence of and the estimate

Let , we have (2.9).

For , we define the following norm:

Suppose that , we have

Corollary 2.5.

For the real number ( fixed) and the integer , where is as in Lemma 2.4, there exists such that for any


From Lemma 2.4,
is a bijective continuous mapping, then is a closed operator. It implies that is also a closed operator. By Banach's closed graph theorem, is continuous. So for any we have

Definition 2.6.

Let be a Banach space. A family of infinitesimal generators of semigroups on is called stable if there are constants and (called the stability constants) such that

for every finite sequence , .

Lemma 2.7.

For , let be the infinitesimal generators of semigroups on the . The family of generators is stable if and only if there are constants and such that

for any finite sequence ,

Lemma 2.8.

Let be a stable family of infinitesimal generators of semigroups on the space such that is independent of and for every , is continuously differentiable in . If , then

has a unique classical solution such that

The proofs of Lemmas 2.7 and 2.8 are in [6]. The straightforward application of the semigroup theory to the system (1.3) gives the following proposition.

Proposition 2.9.

Given and , then there exists one and only one solution of (1.3) such that .


Let . For given , is an infinitesimal generator of semigroups on . For any , it is easy to know that
Then for any , we have
For any finite sequence and any , ,

where . From Lemma 2.4, for any , . Then by Lemma 2.7, is a stable family. Obviously, is continuously differentiable in . So Proposition 2.9 follows from Lemma 2.8.

From Proposition 2.9, we obtain the existence of solutions to the problem (1.1).

Theorem 2.10.

Given and , then there exists one and only one solution of (1.1) such that


Let , . By Proposition 2.9, there exists a solution of problem (1.3) such that . Let denote the forgoing three components of , then is the solution of problem (1.1) and satisfies (2.27).

3. Regularity of Solutions for the Exterior Problem

First, we show the energy inequalities for our problem. These inequalities play an important role in the proof of the regularity of solutions.

Proposition 3.1.

Suppose that
is a solution of problem (1.1) and that , then for any given , we have

where is a constant which depends on .


Put , then and satisfies
where , . Obviously,
By (2.4),
Applying Gronwall's inequality, we get
Without loss of generality, we assume that . Then we see
Applying (3.7) for , we get
By (2.17) and (2.2),
Also we have
and for all ,
Inserting these estimates to the above inequality, we get
An application of Gronwall's inequality implies

This completes the proof of (3.2).

Theorem 3.2.

For , suppose that , , , and
If the compatibility conditions of order are satisfied, then problem (1.1) has a solution such that


At first we prove
Let and . We define by

then ,

We consider the following problem:

here .

From (3.21),
By (3.2), we have
This implies that converges to some in . Set
then tends to in . The passage to the limit of (3.21) shows
Taking account of the definition of , we see

Therefore is the solution of problem (1.1) and satisfies (3.19).

We now prove (3.18) by induction. When , (3.18) follows from (3.2). For , suppose that (3.18) holds for . We show that it still holds for .

Applying (3.2) to (3.27), we conclude from the inductive hypothesis that
In a similar way, we can obtain
Set , then is the solution of (1.3) and
then by (2.17) (taking ), we see
Differentiation of (3.33) with respect to gives
and by the above result ,
from which it follows that
Repeating this process, we get
Using this, we see the right-hand side of (3.33) , then by (2.17) (taking )
This assures that the right-hand side of (3.35) , then
Repeating this process, we get
Step by step, finally, we get

and (3.18).

4. Application to Elastrodynamics

It is well known that the displacement of an isotropic, homogeneous, hyperelastic material without the action of external force satisfies the following hyperbolic system (cf. [4, 5]):
where , and , are given by the Lamé constants , :

We assume that ,

From [5], system (4.1) can be written as

where stands for the elastic tensor.

The system (4.3) is the special case of the system (1.1). So by the existence Theorem 3.2, we derive the existence of solutions for the initial-boundary problem to the elastrodynamic system (4.3) outside a domain.



Projects 10626046 supported by NSFC and 20070410487 supported by China Postdoctoral Science Foundation. The authors would like to thank Professor Tatsien Li and Professor Tiehu Qin for helpful discussions and suggestions.

Authors’ Affiliations

School of Mathematics and Information, Ludong University


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© J. Xin and X. Sha. 2009

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